Surface Area of Rectangular Prism: Express in Terms of X (3×5×X)

Q: Why do we multiply by 2 in the surface area formula?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair has the same area (lw, lh, and wh), so we calculate the area of each unique face and multiply by 2.

Q: How do I remember which dimensions to multiply together?

Think of the three different ways to view the prism: length × width (top/bottom), length × height (front/back), and width × height (left/right sides). Each pair gives you one term in the formula.

Q: What if X equals a specific number like 4?

Simply substitute! If X = 4, then $ 16X + 30 = 16(4) + 30 = 64 + 30 = 94 $ square units. The algebraic expression lets you find surface area for any value of X.

Q: Can I use different letters instead of X?

Absolutely! Variables can be any letter. Whether it's X, t, n, or even d for depth, the process stays the same. Just substitute your variable into the surface area formula.

Q: How do I check if 16X + 30 is correct?

Work backwards: divide by 2 to get $ 8X + 15 $, then see if it equals $ (X)(5) + (X)(3) + (5)(3) = 5X + 3X + 15 = 8X + 15 $ ✓

Surface Area Formula with Variable Dimensions

Express the surface area of the rectangular prism in terms of X using the given data.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the surface area of the box

00:06 We'll use the formula for calculating rectangle area (side times side)

00:13 Calculate the area of each face

00:40 Since the box has 6 faces (2 of each type)

00:44 To calculate the surface area, multiply each area by 2 and sum

01:15 Group the terms

01:19 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Express the surface area of the rectangular prism in terms of X using the given data.

Step-by-step solution

To find the surface area of the rectangular prism in terms of $X$ , follow these steps:

Step 1: Identify the dimensions:
- Length $l = X$
- Width $w = 5$
- Height $h = 3$
Step 2: Use the surface area formula for a rectangular prism: $S = 2(lw + lh + wh)$
Step 3: Substitute the given values: $S = 2(X \cdot 5 + X \cdot 3 + 5 \cdot 3)$
Step 4: Simplify the expression: $S = 2(5X + 3X + 15)$ $S = 2(8X + 15)$ $S = 16X + 30$
Step 5: Therefore, the surface area of the rectangular prism expressed in terms of $X$ is $16X + 30$ .

This matches with choice 1.

Thus, the solution to the problem is $16X + 30$ .

Final Answer

16X+30

Key Points to Remember

Essential concepts to master this topic

Formula: Surface area = 2(lw + lh + wh) for rectangular prisms
Technique: Substitute X·5 + X·3 + 5·3 = 5X + 3X + 15
Check: Factor out 2(8X + 15) = 16X + 30 matches dimensions ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 2 in the surface area formula
Don't calculate just lw + lh + wh = 8X + 15! This gives only half the surface area because each face appears twice on a rectangular prism. Always use the complete formula S = 2(lw + lh + wh) to account for all six faces.

Practice Quiz

Test your knowledge with interactive questions

A cuboid is shown below:

What is the surface area of the cuboid?

FAQ

Everything you need to know about this question

Why do we multiply by 2 in the surface area formula?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair has the same area (lw, lh, and wh), so we calculate the area of each unique face and multiply by 2.

How do I remember which dimensions to multiply together?

Think of the three different ways to view the prism: length × width (top/bottom), length × height (front/back), and width × height (left/right sides). Each pair gives you one term in the formula.

What if X equals a specific number like 4?

Simply substitute! If X = 4, then $16X + 30 = 16(4) + 30 = 64 + 30 = 94$ square units. The algebraic expression lets you find surface area for any value of X.

Can I use different letters instead of X?

Absolutely! Variables can be any letter. Whether it's X, t, n, or even d for depth, the process stays the same. Just substitute your variable into the surface area formula.

How do I check if 16X + 30 is correct?

Work backwards: divide by 2 to get $8X + 15$ , then see if it equals $(X)(5) + (X)(3) + (5)(3) = 5X + 3X + 15 = 8X + 15$ ✓

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